Theorem itgoval 37731

Description: Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Assertion

Ref	Expression
itgoval	|- ( S C_ CC -> (IntgOver ` S ) = { x e. CC | E. p e. (Poly ` S ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } )

Distinct variable group:   x, S, p

Proof of Theorem itgoval

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnex 10017	. . 3 |- CC e. _V
2	1	elpw2 4828	. 2 |- ( S e. ~P CC <-> S C_ CC )
3		fveq2 6191	. . . . 5 |- ( s = S -> (Poly ` s ) = (Poly ` S ) )
4	3	rexeqdv 3145	. . . 4 |- ( s = S -> ( E. p e. (Poly ` s ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) <-> E. p e. (Poly ` S ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) ) )
5	4	rabbidv 3189	. . 3 |- ( s = S -> { x e. CC | E. p e. (Poly ` s ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } = { x e. CC | E. p e. (Poly ` S ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } )
6		df-itgo 37729	. . 3 |- IntgOver = ( s e. ~P CC |-> { x e. CC | E. p e. (Poly ` s ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } )
7	1	rabex 4813	. . 3 |- { x e. CC | E. p e. (Poly ` S ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } e. _V
8	5, 6, 7	fvmpt 6282	. 2 |- ( S e. ~P CC -> (IntgOver ` S ) = { x e. CC | E. p e. (Poly ` S ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } )
9	2, 8	sylbir 225	1 |- ( S C_ CC -> (IntgOver ` S ) = { x e. CC | E. p e. (Poly ` S ) ( ( p ` x ) = 0 /\ ( (coeff ` p ) ` (deg ` p ) ) = 1 ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   ~Pcpw 4158   ` cfv 5888   CCcc 9934   0cc0 9936   1c1 9937  Polycply 23940  coeffccoe 23942  degcdgr 23943  IntgOvercitgo 37727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-itgo 37729

This theorem is referenced by:  aaitgo  37732  itgoss  37733  itgocn  37734